Metamath Proof Explorer

Theorem cdlemg1cex

Description: Any translation is one of our F s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf ? (Contributed by NM, 17-Apr-2013)

Ref Expression
Hypotheses cdlemg1c.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg1c.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg1c.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg1c.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg1cex ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝑇 ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg1c.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg1c.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 cdlemg1c.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 cdlemg1c.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 1 2 3 4 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑝 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑝 ) ≤ 𝑊 ) )
6 5 3expa ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑝 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑝 ) ≤ 𝑊 ) )
7 6 simpld ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑝 ) ∈ 𝐴 )
8 simprr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) → ¬ 𝑝 ≤ 𝑊 )
9 6 simprd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) → ¬ ( 𝐹 ‘ 𝑝 ) ≤ 𝑊 )
10 simpll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) → ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) )
12 simplr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
13 1 2 3 4 cdlemeiota ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑝 ) ) )
14 10 11 12 13 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) → 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑝 ) ) )
15 breq1 ⊢ ( 𝑞 = ( 𝐹 ‘ 𝑝 ) → ( 𝑞 ≤ 𝑊 ↔ ( 𝐹 ‘ 𝑝 ) ≤ 𝑊 ) )
16 15 notbid ⊢ ( 𝑞 = ( 𝐹 ‘ 𝑝 ) → ( ¬ 𝑞 ≤ 𝑊 ↔ ¬ ( 𝐹 ‘ 𝑝 ) ≤ 𝑊 ) )
17 eqeq2 ⊢ ( 𝑞 = ( 𝐹 ‘ 𝑝 ) → ( ( 𝑓 ‘ 𝑝 ) = 𝑞 ↔ ( 𝑓 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑝 ) ) )
18 17 riotabidv ⊢ ( 𝑞 = ( 𝐹 ‘ 𝑝 ) → ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑝 ) ) )
19 18 eqeq2d ⊢ ( 𝑞 = ( 𝐹 ‘ 𝑝 ) → ( 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ↔ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑝 ) ) ) )
20 16 19 3anbi23d ⊢ ( 𝑞 = ( 𝐹 ‘ 𝑝 ) → ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ↔ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ ( 𝐹 ‘ 𝑝 ) ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑝 ) ) ) ) )
21 20 rspcev ⊢ ( ( ( 𝐹 ‘ 𝑝 ) ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ ( 𝐹 ‘ 𝑝 ) ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑝 ) ) ) ) → ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) )
22 7 8 9 14 21 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) → ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) )
23 1 2 3 lhpexnle ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑝 ∈ 𝐴 ¬ 𝑝 ≤ 𝑊 )
24 23 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ∃ 𝑝 ∈ 𝐴 ¬ 𝑝 ≤ 𝑊 )
25 22 24 reximddv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) )
26 25 ex ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝑇 → ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) )
27 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
28 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) → 𝑝 ∈ 𝐴 )
29 simp31 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) → ¬ 𝑝 ≤ 𝑊 )
30 28 29 jca ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) → ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) )
31 simp2r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) → 𝑞 ∈ 𝐴 )
32 simp32 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) → ¬ 𝑞 ≤ 𝑊 )
33 31 32 jca ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) → ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) )
34 simp33 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) → 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) )
35 1 2 3 4 cdlemg1ci2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) → 𝐹 ∈ 𝑇 )
36 27 30 33 34 35 syl31anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) → 𝐹 ∈ 𝑇 )
37 36 3exp ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) → ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) → 𝐹 ∈ 𝑇 ) ) )
38 37 rexlimdvv ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) → 𝐹 ∈ 𝑇 ) )
39 26 38 impbid ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝑇 ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) )